02 / ACCELERATION
MThrough the sound barrier
Hover here to move the shared probe. The dotted line marks M = 1.
COMPRESSIBLE FLOW
Nozzle & Choked FlowShape the passage. Follow the pressure.
QUASI-1D MODELSteady states · live exploration
AXISYMMETRIC SECTION · RADIAL SCALE ENLARGEDLength 1 m
Mach number · M
0123
Drag cyan handles · move over the nozzle to inspect
Solving…
Preparing the nozzle state.
Mass flow——
Throat Mach——
Normal shock——
Total-pressure loss—Across the internal shock
Move over the nozzle or drag the probe slider.
Each edit calculates a new steady state. Flow markers are slowed for visibility; they indicate axial motion, not molecular density. The coloured section contains cross-section averages, not a 2D CFD field.
03 / THE SHOCK SIGNATURE
A jump, and a loss
p / p₀T / T₀Local total p / p₀
Across a normal shock, static pressure rises while total pressure falls.
04 / THE LIMIT
Flow reaches a ceiling
Click this curve to set back pressure. The plateau holds for the current throat area and reservoir conditions.
Checking conservation…
One throat. Two branches.
The same area ratio can admit two Mach numbers. Back pressure selects the complete flow pattern; geometry alone does not choose it.
Choked does not mean stopped.
Once the throat is sonic, further lowering back pressure cannot increase mass flow at fixed inlet total conditions and throat area. The downstream flow can still change.
Keep the boundary visible.
An overexpanded or underexpanded jet adjusts outside the exit. This model reports the pressure mismatch and does not invent an external shock-cell pattern.
Model, equations & verification
A steady compressible-flow model
Calorically perfect air flows through a smooth axisymmetric passage. Properties are uniform over each cross-section. Wall friction, heat transfer and boundary-layer separation are omitted. A fitted normal shock is the only internal irreversible process. Large contour slopes can make the quasi-one-dimensional assumption less representative.
The solver enforces ṁ = ρuA, h₀ = cₚT + u²/2, the isentropic area–Mach relation on each smooth branch, and mass, momentum and energy conservation across a normal shock. Upstream reservoir p₀ and T₀ are held fixed. The inlet section is not the stationary reservoir.
A/A* = M⁻¹ [2/(γ+1) · (1+(γ−1)M²/2)](γ+1)/(2(γ−1))
How the shock position is found
For a choked convergent–divergent nozzle, the solver tests a shock location, calculates the downstream total-pressure loss and subsonic branch, then adjusts the location until exit static pressure matches back pressure. At lower back pressures the interior is supersonic to the exit, and external adjustment is outside this model.
The just-choked back-pressure threshold depends on the exit/throat area ratio. The familiar 0.5283 ratio is p*/p₀ for γ = 1.4; it is the critical back-pressure ratio for the convergent-only case, not every convergent–divergent nozzle.
Each edit is a new equilibrium solution; a moving shock line during adjustment is not a time-accurate shock transient. The displayed graph samples are not a CFD mesh. Formula landmarks, branch inversion, conservation, regime transitions and an input-envelope sweep are checked separately. No experimental accuracy or actual-engine suitability is claimed.
Teaching basis: compressible-flow lecture, pages 9–22; original diagrams created for this lab. Method references: NASA · isentropic flow · NASA · normal shocks · NASA · choking · Virginia Tech · nozzle regimes.